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Note that countable subadditivity, when combined with the empty set axiom, gives as a corollary the finite subadditivity property m∗ (E1 ∪ . . ∪ Ek ) ≤ m∗ (E1 ) + . . + m∗ (Ek ) for any k ≥ 0. These subadditivity properties will be useful in establishing upper bounds on Lebesgue outer measure. Establishing lower bounds will often be a bit trickier. 4. Later on in this text, when we study abstract measure theory on a general set X, we will define the concept of an outer measure on X, which is an assigment E → m∗ (E) of element of [0, +∞] to arbitrary subsets E of a space X that obeys the above three axioms of the empty set, monotonicity, and countable subadditivity; thus Lebesgue outer measure is a model example of an abstract outer measure.

For simplicity we will only discuss the classical one-dimensional Riemann integral on an interval [a, b], though one can extend the Riemann theory without much difficulty to higher-dimensional integrals on Jordan measurable sets. 5 (Riemann integrability). Let [a, b] be an interval of positive length, and f : [a, b] → R be a function. A tagged partition P = ((x0 , x1 , . . , xn ), (x∗1 , . . , x∗n )) of [a, b] is a finite sequence of real numbers a = x0 < x1 < . . < xn = b, together with additional numbers xi−1 ≤ x∗i ≤ xi for each i = 1, .

However, observe that given any r > 0, one can always partition a large box Bn into a finite number of smaller boxes, each of which has diameter11 at most r, with the total volume of these sub-boxes equal to the volume of the original box. Applying this observation to each of the boxes Bn , we see that given any r > 0, we may assume without loss of generality that the boxes B1 , B2 , . . covering E ∪F have diameter at most r. In particular, we may assume that all such boxes have diameter strictly less than dist(E, F ).

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